MA 15200 - Purdue University



MA 15200 Lesson 28 Section 4.2

Remember the following information about inverse functions.

1. In order for a function to have an inverse, it must be one-to-one and pass a horizontal line test.

2. The inverse function can be found by interchanging x and y in the function’s equation and solving for y.

3. If [pic]. The domain of f is the range of [pic] and the range of f is the domain of [pic].

4. The compositions [pic] both equal x.

5. The graph of [pic] is the reflection of the graph of f about the line [pic].

Because an exponential function is 1-1 and passes the horizontal line test, it has an inverse. This inverse is called a logarithmic function.

I Logarithmic Functions

According to point 2 above, we interchange the x and y and solve for y to find the equation of an inverse function.

[pic] exponential function

[pic] How do we solve for y? There is no way to do this.

Therefore a new notation needs to be used to represent an inverse of an exponential function, the logarithmic function.

Definition of Logarithmic Function

[pic]

The function [pic] is the logarithmic function with base b.

The equation [pic] is called the logarithmic form and the equation [pic] is called the exponential form. The value of y in either form is called a logarithm. Note: The logarithm is an exponent.

Exponential Form Logarithmic Form

[pic] [pic]

argument base exponent exponent base argument

Ex 1: Convert each exponential form to logarithmic form and each logarithmic form to exponential form.

[pic]

[pic]

II Finding logarithms

Remember: A logarithm is an exponent.

Ex 2: Find each logarithm.

[pic]

[pic]

[pic]

III Basic Logarithmic Properties

1. [pic] Since the first power of any base equals that base, this is reasonable.

2. [pic] Since any base to the zero power is 1, this is reasonable.

The exponential function [pic] and the logarithmic function [pic] are inverses.

[pic]

This leads to 2 more basic logarithmic properties.

3. [pic] This is a composition function where [pic]. [pic] (the exponent)

4. [pic] This is a composition function where [pic]. [pic](the number or argument)

Ex 3: Simplify using the basic properties of logarithms.

[pic]

Ex 4: Simplify, if possible.

[pic]

IV Graphs of Logarithmic Functions

Below is a graph of [pic].

[pic]

If you imagine the line y = x, you can see the symmetry about that line.

Below are both graphs on the same coordinate system along with y = x.

[pic]

Characteristics of a logarithmic Graph:

The inverse of this function, [pic], has a graph with the following characteristics.

1. The x-intercept is (1, 0).

2. The graph still is increasing if b > 1, decreasing if 0 < b < 1.

3. The domain is all positive numbers, so the graph is to the right of the y-axis. (The range is all real numbers.)

4. The y-axis is an asymptote.

V Common Logarithms

A logarithmic function with base 10 is called the common logarithmic function. Such a function is usually written without the 10 as the base.

[pic]

A calculator with a key can approximate common logarithms.

Put the number (argument) in the calculator, press the common log key.

Ex 5: Find each common logarithm without a calculator.

[pic]

Ex 6: Use a calculator to approximate each common logarithm. Round to 4 decimal places.

[pic]

Using the basic properties with base 10, we get the following properties.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

VI Natural Logarithms

A logarithm function with base e is called the natural logarithmic function. Such a function is usually written using ln rather than log and no base shown.

[pic]

A calculator with a key can approximate natural logarithms.

Put the number (argument) in the calculator, press the natural log key.

Ex 7: Use a calculator to approximate each natural logarithm. Round to 4 decimal places.

[pic]

Using the basic properties with base e, we get the following properties.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

VII Modeling with logarithmic functions

The function [pic] gives the percentage of adult height attained by a boy who is x years old.

Ex 8: Approximately what percentage of his adult height has a boy of age 11 acheived?

(Notice: This model uses a common log.) Round to the nearest tenth of a percent.

The function [pic] models the temperature increase in degrees Fahrenheit after x minutes in an enclosed vehicle when the outside temperature is from 72° to 96°.

Ex 9: Use the function above to approximate the temperature increase after 45 minutes. Round to the nearest tenth of a degree.

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[pic]

Remember that bases must be positive and the argument values (the numbers) must be positive.

(2, 1)

(1, 2)

log

ln

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